Worst-Case Control Mechanism for Approximate Symbolic Analysis ZDENEK KOLKA 1) , VIERA BIOLKOVA 1) , DALIBOR BIOLEK 2) 1) Department of Radioelectronics, Brno University of Technology 2) Department of Electrical Engineering, University of Defence, Brno CZECH REPUBLIC kolka@feec.vutbr.cz, biolkova@feec.vutbr.cz, dalibor.biolek@unob.cz Abstract: The paper deals with a control mechanism for approximate symbolic analysis of the Simplification- Before-Generation class that uses the worst case analysis to incorporate component variations into the ranking mechanism. The traditional approach consists in the evaluation of errors caused by simplification only for the nominal parameters of network components. The proposed procedure uses a fast algorithm for computing the sensitivities needed for the vertex-based worst-case analysis. An advantage of the proposed approach is that the validity of approximated expression is checked over the tolerance interval of component parameters. Key-Words: Symbolic analysis, Simplification, Error prediction, Worst-case analysis, Linear circuits; CAD tools 1 Introduction The applicability of exact symbolic analysis in the frequency domain is constrained to relatively small linear circuits, as the size of the resulting expression grows exponentially with the number of nodes and components. If we restrict the range of frequency and network parameters, the majority of symbolic terms can be removed from large expressions without any significant numerical error [1]. Simplification methods can be divided into three classes according to the stage of analysis at which the simplification is performed: Simplification Before Generation (SBG), Simplification During Generation (SDG), and Simplification After Generation (SAG) [2]. The SAG methods are algorithmically simple but very expensive in terms of computation and storage. Methods of the SDG type are rather complex and may have problems with the interpretability of resulting expressions [3]. The SBG methods directly modify the network model. Thus the simplification is inherently more appropriate and intuitive from the point of view of the expression interpretability [3]. Several SBG methods have been proposed so far. They operate with network matrices [2], [4] or graphs [5], [6], [7]. The simplification procedure is essentially the same for all the methods. First, all the prospective simplifications (circuit element removal, matrix or graph modification, etc.) are ranked numerically according to the error their application would cause. One or more operations with the lowest error are actually performed and the numerical solution is updated. The procedure is repeated until the maximum error is reached. All computations of errors should be done numerically, as neither the full nor the simplified functions are available in the symbolic form during the simplification process. In all cases, an adequate error mechanism is needed to control which term or parameter can be deleted without exceeding some prescribed maximum magnitude and phase errors [1]. Theoretically, the maximum error should be guaranteed on some continuous frequency interval and for network parameters having any values from their tolerance ranges. The traditional approach consists in the evaluation of errors on a set of frequency samples and for nominal parameters of network components [2], [4]. For a low number of frequency samples there is a danger of exceeding the maximum error between the samples. A method in [6] uses interval analysis techniques to detect such situations. From the point of view of network parameters, the ranking criterion should ideally be based on the Worst-Case analysis. Applicable methods include the interval analysis and the sensitivity-based vertex analysis [8]. A simple application of the interval analysis [9] overestimates the resulting uncertainty due to the intractable interval expansion caused by dependence among interval operands. A tight- interval analysis for linear circuits was proposed, for example, in [10] or [11]. However, these methods were designed to estimate a perturbed solution of the linear system Ax = b and cannot be used directly for ranking, which uses the ratio of two perturbed solutions (see Section 2). Under the assumption of monotonicity the vertex analysis provides that the worst case parameter sets are located at the vertices of parameter space. The method is very fast if the right vertex is identified Mathematical Models and Methods in Modern Science ISBN: 978-1-61804-055-8 160